x Turn on thread page Beta
 You are Here: Home >< Maths

# Curl of a cross product watch

1. So I'm trying to derive the identity for the curl of (u x v) - so I've worked out u x v then taken the curl of that in the usual ways (i.e. determinants) - but the resulting expression is a horrible horrible mess and only gets worse when you start to invoke the product rule on each of the terms.

Final answer should be

curl (u x v) = (v.del)u - (u.del)v + u(div v) - v(div u)

I'm not crazy about trying to spot how the terms in the messy expression I've got collapse into this lot...

Two questions, then:

1. Is there a neater way of doing this (perhaps using index notation/summation convention - its fairly easy to get expressions in index notation for div and grad as far as I can tell, but curl isnt as easy to do this with...)

2. More fundamentally, why doesnt a x (b x c) = b(a.c) - c(a.b) work here? Something to do with the fact that del is a vector operator rather than a vector maybe??

Thanks
2. With a question that looks as horrible as you say I would evaluate both sides and find an equivalence somewhere in the middle.

I think you'l find that the identity you are considering is more because it is a differential operator than anything else.
3. (Original post by -G-a-v-)
So I'm trying to derive the identity for the curl of (u x v) - so I've worked out u x v then taken the curl of that in the usual ways (i.e. determinants) - but the resulting expression is a horrible horrible mess and only gets worse when you start to invoke the product rule on each of the terms.

Final answer should be

curl (u x v) = (v.del)u - (u.del)v + u(div v) - v(div u)

I'm not crazy about trying to spot how the terms in the messy expression I've got collapse into this lot...

Two questions, then:

1. Is there a neater way of doing this (perhaps using index notation/summation convention - its fairly easy to get expressions in index notation for div and grad as far as I can tell, but curl isnt as easy to do this with...)

2. More fundamentally, why doesnt a x (b x c) = b(a.c) - c(a.b) work here? Something to do with the fact that del is a vector operator rather than a vector maybe??

Thanks
i don't know how to use :atex so this may be a mess.!

let eijk be the Levi-Civita symbol

and fi be the ith unit vector then

eijk = fi . (fj x fk) (check this yourself)

so eijk fi = (fi . (fj x fk)) fi = fj x fk

so for any cont diff vector function G

curl G = eijk fi d(Gk)/dxj (curl in summation convention)

=fj x fk d(Gk)/dxj (Gk is the kth component of G)

=fj x d(G)/dxj

so curl(u x v) = fj x d(u x v)/dxj

Then bring fj inside the derivitive as constant then expand
(all the derivities are partials obviously)

it requires 7 lines of working and use summation convention

Hope this helps

Turn on thread page Beta
TSR Support Team

We have a brilliant team of more than 60 Support Team members looking after discussions on The Student Room, helping to make it a fun, safe and useful place to hang out.

This forum is supported by:
Updated: February 7, 2010
Today on TSR

### Any tips for freshers...

who are introverts?

### More snow?!

Discussions on TSR

• Latest
Poll
Useful resources

## Make your revision easier

### Maths Forum posting guidelines

Not sure where to post? Read the updated guidelines here

### How to use LaTex

Writing equations the easy way

### Study habits of A* students

Top tips from students who have already aced their exams

Can you help? Study help unanswered threads

## Groups associated with this forum:

View associated groups
Discussions on TSR

• Latest

The Student Room, Get Revising and Marked by Teachers are trading names of The Student Room Group Ltd.

Register Number: 04666380 (England and Wales), VAT No. 806 8067 22 Registered Office: International House, Queens Road, Brighton, BN1 3XE